Method for determination of apparent resistivities of anisotropic reservoirs

ABSTRACT

Shoulder corrections are applied to measurements obtained from a multi-component electromagnetic logging tool. An anisotropic resistivity model is obtained using the shoulder corrected data. The process is iterated until a good match is obtained between the shoulder corrected data and the model output.

CROSS-REFERENCES TO RELATED APPLICATIONS

[0001] This application claims priority from U.S. Provisional PatentApplication Ser. No. 60/164,055 filed on Nov. 8, 1999.

BACKGROUND OF THE INVENTION

[0002] 1. Field of the Invention

[0003] The invention is related generally to the field of interpretationof measurements made by well logging instruments for the purpose ofdetermining the properties of earth formations. More specifically, theinvention is related to methods for correcting measurements made bymulti-component induction or propagation sensors for shoulder bed andborehole effects.

[0004] 2. Background of the Art

[0005] Electromagnetic induction and wave propagation logging tools arecommonly used for determination of electrical properties of formationssurrounding a borehole. These logging tools give measurements ofapparent resistivity (or conductivity) of the formation that whenproperly interpreted are diagnostic of the petrophysical properties ofthe formation and the fluids therein.

[0006] The physical principles of electromagnetic induction resistivitywell logging are described, for example, in, H. G. Doll, Introduction toInduction Logging and Application to Logging of Wells Drilled with OilBased Mud, Journal of Petroleum Technology, vol. 1, p. 148, Society ofPetroleum Engineers, Richardson Tex. (1949). Many improvements andmodifications to electromagnetic induction resistivity instruments havebeen devised since publication of the Doll reference, supra. Examples ofsuch modifications and improvements can be found, for example, in U.S.Pat. No. 4,837,517; U.S. Pat. No. 5,157,605 issued to Chandler et al,and U.S. Pat. No. 5,452,761 issued to Beard et al.

[0007] A limitation to the electromagnetic induction resistivity welllogging instruments known in the art is that they typically includetransmitter coils and receiver coils wound so that the magnetic momentsof these coils are substantially parallel only to the axis of theinstrument. Eddy currents are induced in the earth formations from themagnetic field generated by the transmitter coil, and in the inductioninstruments known in the art these eddy currents tend to flow in groundloops which are substantially perpendicular to the axis of theinstrument. Voltages are then induced in the receiver coils related tothe magnitude of the eddy currents. Certain earth formations, however,consist of thin layers of electrically conductive materials interleavedwith thin layers of substantially non-conductive material. The responseof the typical electromagnetic induction resistivity well logginginstrument will be largely dependent on the conductivity of theconductive layers when the layers are substantially parallel to the flowpath of the eddy currents. The substantially non-conductive layers willcontribute only a small amount to the overall response of the instrumentand therefore their presence will typically be masked by the presence ofthe conductive layers. The non-conductive layers, however, are the oneswhich are typically hydrocarbon-bearing and are of the most interest tothe instrument user. Some earth formations which might be of commercialinterest therefore may be overlooked by interpreting a well log madeusing the electromagnetic induction resistivity well logging instrumentsknown in the art.

[0008] U.S. Pat. No. 5,999,883 issued to Gupta et al, (the “Guptapatent”), the contents of which are fully incorporated here byreference, discloses a method for determination of the horizontal andvertical conductivity of anisotropic earth formations. Electromagneticinduction signals induced by induction transmitters oriented along threemutually orthogonal axes are measured. One of the mutually orthogonalaxes is substantially parallel to a logging instrument axis. Theelectromagnetic induction signals are measured using first receiverseach having a magnetic moment parallel to one of the orthogonal axes andusing second receivers each having a magnetic moment perpendicular to aone of the orthogonal axes which is also perpendicular to the instrumentaxis. A relative angle of rotation of the perpendicular one of theorthogonal axes is calculated from the receiver signals measuredperpendicular to the instrument axis. An intermediate measurement tensoris calculated by rotating magnitudes of the receiver signals through anegative of the angle of rotation. A relative angle of inclination ofone of the orthogonal axes which is parallel to the axis of theinstrument is calculated, from the rotated magnitudes, with respect to adirection of the vertical conductivity. The rotated magnitudes arerotated through a negative of the angle of inclination. Horizontalconductivity is calculated from the magnitudes of the receiver signalsafter the second step of rotation. An anisotropy parameter is calculatedfrom the receiver signal magnitudes after the second step of rotation.Vertical conductivity is calculated from the horizontal conductivity andthe anisotropy parameter.

[0009] Shoulder bed corrections related to the effect of formationsabove and below the depth being evaluated also have to be applied to thedata. Methods for making these corrections to data acquired withconventional logging tools are well known in the art.

[0010] For example, U.S. Pat. No. 5,446,654 to Chemali teaches theconversion of a resistivity log as a function of well depth into arectangularized curve so that the interfaces of the adjacent strata arelocated, and by a suitable number of iterations, a correction factor isapplied. The corrected rectangular log is obtained with a correctioncoefficient computed at each depth. For each computation, the impact ofall the strata within a specified depth window is considered, whilestrata beyond that window are simplified by representing the stratabeyond the window with single equivalent bed values to reduce the numberof computations required. This then provides a resistivity log which issubstantially free of shoulder bed effect.

[0011] The method of U.S. Pat. No. 5,867,806 to Strickland et al.selects one or more control depths at one or more locations of each of aplurality of detected beds in the formation. The method then estimatesthe resistivity of each bed only at the selected control depths toproduce an estimated resistivity of the beds. The method then computes asimulated log value at each control depth using a current estimate ofthe resistivity of the beds. The computed simulated log is then comparedto the actual log data at each control depth, and the resistivity ofeach bed is adjusted using the difference between the actual andsimulated values at the control depths. The above method iterativelyrepeats a plurality of times until the simulated log substantiallymatches the actual log at the control depths.

[0012] There is a need for a method of shoulder bed correction ofmulticomponent resistivity data so as to improve the estimatedhorizontal and vertical formation resistivities obtained by inversion ofthe shoulder-bed corrected data. Such a method should preferablycomputationally efficient so as to provide the necessary corrections atthe wellsite and in real time. The present invention satisfies thisneed.

SUMMARY OF THE INVENTION

[0013] The present invention is a method for determining an applyingshoulder bed corrections to logging measurements made with a transverseinduction logging tool. Layer boundaries are determined from themeasurements. These are combined with horizontal and verticalresistivities obtained by a whole space anisotropic inversion to give alayered model. Preferably, a Lanczos iterative procedure is used for theinversion. The shoulder bed correction for each layer is derived basedupon a difference between a 1-D synthetic response of the model and awhole space response of the model at that layer. The shoulder bedcorrection is applied to the data and the inversion procedure isrepeated. This procedure is repeated in an iterative manner until adifference between the shoulder bed corrected measurements at the centerof each of the layers and a synthetic response to a whole space model atthe center of each of the layers is below a predetermined threshold.

BRIEF DESCRIPTION OF THE FIGURES

[0014]FIG. 1 shows an induction instrument disposed in a wellborepenetrating earth formations.

[0015]FIG. 2 shows the arrangement of transmitter and receiver coils ina preferred embodiment of the present invention marketed under the name3DExplorer™

[0016]FIG. 3 shows examples of the response of some of the coils of theinstrument of FIG. 3 to an anisotropic earth.

[0017]FIG. 4 shows a flow chart of a preferred embodiment of the presentinvention for applying shoulder bed corrections to data from atransverse induction logging tool.

DETAILED DESCRIPTION OF THE INVENTION

[0018] Referring now to FIG. 1, an electromagnetic induction welllogging instrument 10 is shown disposed in a wellbore 2 drilled throughearth formations. The earth formations are shown generally at 4. Theinstrument 10 can be lowered into and withdrawn from the wellbore 2 bymeans of an armored electrical cable 6 or similar conveyance known inthe art. The instrument 10 can be assembled from three subsections: anauxiliary electronics unit 14 disposed at one end of the instrument 10;a coil mandrel unit 8 attached to the auxiliary electronics unit 14; anda receiver/signal processing/telemetry electronics unit 12 attached tothe other end of the coil mandrel unit 8, this unit 12 typically beingattached to the cable 6.

[0019] The coil mandrel unit 8 includes induction transmitter andreceiver coils, as will be further explained, for inducingelectromagnetic fields in the earth formations 4 and for receivingvoltage signals induced by eddy currents flowing in the earth formations4 as a result of the electromagnetic fields induced therein.

[0020] The auxiliary electronics unit 14 can include a signal generatorand power amplifiers (not shown) to cause alternating currents ofselected frequencies to flow through transmitter coils in the coilmandrel unit 8.

[0021] The receiver/signal processing/telemetry electronics unit 12 caninclude receiver circuits (not shown) for detecting voltages induced inreceiver coils in the coil mandrel unit 8, and circuits for processingthese received voltages (not shown) into signals representative of theconductivities of various layers, shown as 4A through 4F of the earthformations 4. As a matter of convenience the receiver/signalprocessing/telemetry electronics unit 12 can include signal telemetry totransmit the conductivity-related signals to the earth's surface alongthe cable 6 for further processing, or alternatively can store theconductivity related signals in an appropriate recording device (notshown) for processing after the instrument 10 is withdrawn from thewellbore 2.

[0022] Turning now to FIG. 2, the configuration of transmitter andreceiver coils in a preferred embodiment of the 3DExplorer™ inductionlogging instrument of Baker Hughes is disclosed. Three orthogonaltransmitters 101, 103 and 105 that are referred to as the T_(x), T_(z),and T_(y) transmitters are shown (the z-axis is the longitudinal axis ofthe tool). Corresponding to the transmitters 101, 103 and 105 areassociated receivers 107, 109 and 111, referred to as the R_(x), R_(z),and R_(y) receivers, for measuring the corresponding magnetic fieldsH_(xx), H_(zz), and H_(yy). In addition, the receivers 113 and 115measure two cross-components H_(xy), and H_(xz). of the magnetic fieldproduced by the x-component transmitter.

[0023] The problems with logging in anisotropic media are brought outwith reference to FIG. 3. Shown on the left panel is a resisitivitymodel with horizontal and vertical resistivities denoted by 201 a and201 b. The model has three anisotropic intervals 203, 205, 207 where thevertical resisitivity R_(v) is greater than the horizontal resisitivityR_(h). The right panel in FIG. 3 shows the apparent conductivityresponses for the H_(xx) 211 and H_(zz) 215 components. Also shown isthe H_(xx) component 213 if the resisitivity model is isotropic. TheH_(zz) response for an isotropic model is the same as for theanisotropic model.

[0024] From FIG. 3, the following observations may be made about theresisitivity responses for a vertical well in an anisotropic formation.The H_(zz) curve is not responsive to anisotropy in the formation. TheH_(xx) curve is quite complicated and can even reverse sign close tosignificant resisitivity contrasts. It also may have spikes at bedboundaries. Furthermore, the H_(xx) curve is not indicative of theresistive or conductive nature of a bed. Additionally, the H_(xx)response exhibits more skin effect than does the H_(zz) response.

[0025] Turning now to FIG. 4, a flow chart illustrating the principalsteps of the present invention is shown. The first step is obtainingmulti-component induction logging data. This may be done by use of thetool described above with reference to FIGS. 1 and 2 or by any othersuitable instrument. These measurements are inverted using a whole spaceinversion 303.

[0026] In the whole space inversion, first the synthetic tool responsein an isotropic 1-Ω-m whole-space model, i.e., without horizontal orvertical boundaries, is calculated. The synthetic whole-space responseis then compared with measured field data at each logging depth and thehorizontal and vertical resistivities ( R_(h) and R_(v)) are adjusted tomatch the synthetic responses with the measured field responses. Inperforming the match, the borehole inclination (relative to the layers)and the azimuth are required input parameters. The borehole inclinationand azimuth may be obtained using a method such as is taught by U.S.Pat. No. 5,999,883 to Gupta et al., the contents of which are fullyincorporated here by reference.

[0027] The model response depends nonlinearly upon a model parametervector m comprising a plurality of layers each having an associatedvalue of R_(h) and R_(v). $\begin{matrix}{\begin{bmatrix}m_{1} \\m_{2} \\m_{3} \\m_{4} \\\vdots \\m_{{2n} - 1} \\m_{2n}\end{bmatrix} = \begin{bmatrix}R_{h1} \\R_{v1} \\R_{h2} \\R_{v2} \\\vdots \\R_{vn} \\R_{hn}\end{bmatrix}} & (1)\end{matrix}$

[0028] where n is the number of layers in the model.

[0029] The synthetic response d of a model m is related in a nonlinearmanner to the model and the synthetic tool response h calculated aboveby a nonlinear equation of the form

d= f ( m,h )   (2)

[0030] The inversion is performed by an iterative process

m _(k+1)=m _(k)+Δm _(k)   (3)

[0031] where m _(k) is the model parameter vector at the k-th iterationand Δm _(k) is the parameter update vector calculated as

Δm _(k)=(j^(T)j+αI)⁻¹ (J^(T) Δd)   (4)

[0032] In eq. (4), J is the Jacobian or sensitivity matrix of partialderivatives of changes in the data to small changes in the parameter.The difference between the measured and the synthetic data is Δd, α is aregularization parameter and I is the identity matrix.

[0033] In a preferred embodiment of the invention, Eq. (4) is solvedusing a Singular Value Decomposition as taught by Lanczos although anyother method may be used. The inversion process is extremely fastbecause it only involves analytical solutions of the anisotropicresponse as represented by eq. (2).

[0034] The obtained multi-component log data are also input to a bedboundary detection step 305 . U.S. Pat. No. 5,999,884 to Kriegshauser etal. having the same assignee as the present application and the contentsof which are fully incorporated herein by reference, teaches a methodfor estimating axial positions of formation layer boundaries fromtransverse electromagnetic induction signals.

[0035] In the Kriegshauser patent, a first derivative is calculated withrespect to depth of the induction signals. Next, a second derivative ofthe signals is calculated. The second derivative is muted. Layerboundaries are selected at axial positions where the muted secondderivative is non zero, and the first derivative changes sign. Theselected boundaries are thickness filtered to eliminate boundaries whichhave the same axial spacing as the spacing between an inductiontransmitter and receiver used to measure the induction signals, and toeliminate boundaries having a spacing less than an axial resolution ofthe induction signals. The process is repeated using transverseinduction measurements made at another frequency. Layer boundaries thatare common to the two frequency determinations are determined to be thelayer boundaries.

[0036] Alternatively, Kriegshauser discloses a frequency domain methodfor bed boundary determination. The induction signals are transformedinto the spatial frequency domain, and low pass filtered at a band limitabout equal to the axial resolution of the induction signals. Thecentral first derivative of the filtered signals is calculated and thecentral first derivative inverse-transformed back to the spatial domain.Zero crossings of the inverse-transformed first derivative areindicative of formation boundaries.

[0037] The output of the inversion 303 and the output of the bedboundary detection 305 are combined to give an initial 1-D layered modelof the earth. This initial model comprises the layers from 305 andassociated horizontal and vertical resistivities from 303. From theinitial model, the shoulder bed corrections are calculated and appliedto the measurements 309.

[0038] To determine the should bed corrections, the followingsub-process is carried out:

[0039] 1. Calculate D_(1Dver), the 1-D layered synthetic response iscalculated from the initial model using the impulse response describedabove with reference to the whole-space inversion step 303.

[0040] 2. Calculate D_(WS), the whole space response at each layer usingthe horizontal and vertical resistivities for that layer derived in thewhole space inversion 303.

[0041] 3. Calculate the shoulder bed effect ΔD_(SB), the shoulder bedcorrection as

ΔD _(SB) =D _(1Dver) −D _(WS)   (5)

[0042] 4. Apply the corrections AD_(SB) to the measurements to give$\begin{matrix}{H_{meas}^{{corr},{SB}} = {H_{meas} - {\Delta \quad D_{SB}}}} & (6)\end{matrix}$

[0043] After applying the corrections, the whole space inversion iscarried out again 311 The newly derived resistivities are then used torefine the 1-D layered model 313 in a manner similar to that describedabove with respect to 307; in doing so, the bed boundaries derived aboveat 305 are used.

[0044] A check is made to see if the model is acceptable 315. The testfor acceptability is that the whole space response of the layered modelat the middle of each layer should be within some predeterminedthreshold of the measurements shoulder bed corrected measurements.Alternatively, the iterative procedure is carried out for a specifiednumber of iterations. If the model is not acceptable, the processiteratively goes back to 309 and steps 309-313 are repeated until eitherthe model is acceptable or the iterative process has been carried out apredetermined number of times. If the model is acceptable, the processterminates 317 and the model may be used for subsequent petrophysicalprocessing and interpretation. For example, water saturation may bedetermined for the layers in the inverted model from a knowledge of theresistivities of the layers.

[0045] The present invention has been discussed above with respect tomeasurements made by a transverse induction logging tool conveyed on awireline. This is not intended to be a limitation and the method isequally applicable to measurements made using a comparable tool conveyedon a measurement-while-drilling (MWD) assembly or on coiled tubing.

[0046] While the foregoing disclosure is directed to the preferredembodiments of the invention, various modifications will be apparent tothose skilled in the art. It is intended that all variations within thescope and spirit of the appended claims be embraced by the foregoingdisclosure.

What is claimed is:
 1. A method of determining a parameter of interestof subsurface formations containing a sand and a shale surrounding aborehole, the method comprising: (a) conveying an electromagneticlogging tool into the borehole and using at least one electromagnetictransmitter and one electromagnetic receiver on the tool to obtainmeasurements related to a horizontal and vertical resistivity of theformation; (b) obtaining a plurality of layer intervals for thesubsurface; (c) deriving a horizontal and vertical resistivity at aplurality of depths from said measurements using an anisotropic wholespace model; (d) obtaining from the derived horizontal and verticalresistivities a layered model; (e) deriving a shoulder bed correctionsbased upon the layered model and the whole space model; (f) applying theshoulder bed corrections to said measurements to obtain shoulder bedcorrected measurements; (g) determining a difference between saidshoulder bed corrected measurements at the center of each said pluralityof layers to a synthetic response of the whole space model at saidcenters; and (h) iteratively repeating steps (c)-(g) to give an updatedlayered model.
 2. The method of claim 1 wherein said electromagneticlogging tool is conveyed one one of (i) a wireline, (ii) a drillstring,and (iii) coiled tubing.
 3. The method of claim 1 wherein the parameterof interest is at least one of (i) a horizontal resistivity of one ofsaid layers, (ii) a vertical resistivity of one of said layers, (iii) afluid saturation of one of said layers.
 4. The method of claim 1 whereinderiving said horizontal and vertical resistivity at said plurality ofdepths further comprises performing a whole space inversion.
 5. Themethod of claim 4 wherein performing a whole space inversion furthercomprises computing a synthetic tool response.
 6. The method of claim 4wherein performing a whole space inversion further comprises updatingestimates of said horizontal and vertical resistivity using a Lanczosprocedure.
 7. The method of claim 1 wherein obtaining said plurality oflayer intervals further comprises determining a first and a secondderivative with respect to depth of said measurements.
 8. The method ofclaim 1 wherein obtaining said plurality of layer intervals furthercomprises: (i) Fourier transforming said signals into the spatialfrequency domain; (ii) low pass filtering said Fourier transformedsignals at a cutoff about equal to an axial resolution of said inductionsignals; (iii) calculating a central first derivative of said filteredFourier transformed signals; (iv) calculating an inverse Fouriertransform of said central first derivative; (v) selecting roots of saidinverse Fourier transformed central first derivative; and (vi) testinglocalized properties of said inverse Fourier transformed central firstderivative within a selected number of data sample points of said roots,thereby providing indications of formation layer boundaries at axialpositions most likely to be true ones of said formation layerboundaries.
 9. The method of claim 1 wherein deriving shoulder bedcorrections further comprises: (i) determining a 1-D layered syntheticresponse is calculated from the layered model using a synthetic toolresponse; (ii) determining a whole space response using said horizontaland vertical resistivities; and (iii) obtaining said shoulder bedcorrection as a difference between said 1-D layered synthetic responseand said whole space response.
 10. The method of claim 1 wherein step(h) is carried out until said difference is less than a predeterminedthreshold.
 11. The method of claim 1 wherein step (h) is carried out aprespecified number of iterations.